Electromagnetic control of thermal convection of a fluid with strongly temperature-dependent material properties

Abstract

We study a one-dimensional model describing buoyancy-driven laminar steady flow of a glass melt in a closed loop under the influence of a localized electromagnetic (Lorentz) force. The loop is a highly simplified representation of a closed streamline in glass melt flow in a real furnace under the influence of an artificially produced Lorentz force. The model is based on the energy equation for the temperature and the Stokes equation for the velocity distribution inside the loop. We take into account the full nonlinear temperature dependence of the viscosity and the electrical conductivity of the melt. The three-dimensional problem is then reduced to a single nonlinear equation for the cross-section averaged velocity from which the one-dimensional temperature distribution along the loop can be readily obtained. We show that the two-way interaction between the velocity and temperature resulting from the temperature-dependent material properties and Lorentz force leads to the result that the mean velocity as a function of the control parameters is non-unique and involves bifurcations. For some parameters we even observe freezing, which refers to a regime in which the fluid is almost at rest. Our model reveals the role of temperature-dependent viscosity and conductivity in glass melt flows in a pure form that is not visible in full numerical simulations.